A Study of Complex Shapes Using Information Geometry

Abstract

In this paper, a new statistical method is proposed to model patterns emerging in complex systems. It uses several tools of Information Geometry to describe the evolution of various shapes that arise due to internal forces of the system. A framework for shape analysis of 2− dimensional landmark data is introduced in which each landmark is represented by a bivariate Gaussian distribution, reflecting the uncertainty in the landmark’s placement or the natural variability across a population of shapes. The Fisher–Rao metric is considered as a natural measure of the distance between two distributions. It endows the statistical manifold of parameters with a Riemannian metric, thus allowing the interpolation between observed shapes by computing the geodesics between the corresponding densities. Furthermore, the geodesic path connecting two shapes can be used for shape predictions. We also apply the method to study the evolution of the rat skull shape.